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Abstract An interior-point algorithm framework is proposed, analyzed, and tested for solving nonlinearly constrained continuous optimization problems. The main setting of interest is when the objective and inequality constraint functions may be nonlinear and/or nonconvex, and when constraint values and derivatives are tractable to compute, but objective function values and derivatives can only be estimated. The algorithm is intended primarily for a setting that is similar for stochastic-gradient methods for unconstrained optimization, i.e., the setting when stochastic-gradient estimates are available and employed in place of gradients, and when no objective function values (nor estimates of them) are employed. This is achieved by the interior-point framework having a single-loop structure rather than the nested-loop structure that is typical of contemporary interior-point methods. Convergence guarantees for the framework are provided both for deterministic and stochastic settings. Numerical experiments show that the algorithm yields good performance on a large set of test problems. 

Abstract Distributionally Favorable Optimization (DFO) is a framework for decision-making under uncertainty, with applications spanning various fields, including reinforcement learning, online learning, robust statistics, chance-constrained programming, and two-stage stochastic optimization without complete recourse. In contrast to the traditional Distributionally Robust Optimization (DRO) paradigm, DFO presents a unique challenge– the application of the inner infimum operator often fails to retain the convexity. In light of this challenge, we study the tractability and complexity of DFO. We establish sufficient and necessary conditions for determining when DFO problems are tractable (i.e., solvable in polynomial time) or intractable (i.e., not solvable in polynomial time). Despite the typical nonconvex nature of DFO problems, our results show that they are mixed-integer convex programming representable (MICP-R), thereby enabling solutions via standard optimization solvers. Finally, we numerically validate the efficacy of our MICP-R formulations. 

There is a growing need for electricity-system flexibility to maintain real-time balance between energy supply and demand. In this paper, we explore optimal and incentive-compatible scheduling of generators for this purpose. Specifically, we examine a setting wherein each generator has a different operating cost if it is committed in advance (e.g., day- or hour-ahead) as opposed to being reserved as flexible real-time supply. We model an optimal division of generators between advanced commitment and real-time flexible reserves to minimize the expected cost of serving an uncertain demand. Next, we propose an incentive-compatible remuneration scheme with two key properties. First, the remuneration scheme incentivizes generators to reveal their true costs. Second, the scheme aligns generators’ incentives with the market operator’s optimal division of generators between advanced commitment and real-time reserve. We use a simple example to illustrate the market operator’s decision and the remuneration scheme. JEL Classification: C61, D47, D82, L94, Q4